WASHINGTON — The blaring, grinding noise jolted the American diplomat from his bed in a Havana hotel. He moved just a few feet, and there was silence. He climbed back into bed. Inexplicably, the agonizing sound hit him again. It was as if he’d walked through some invisible wall cutting straight through his room.

Soon came the hearing loss, and the speech problems, symptoms both similar and altogether different from others among at least 21 U.S. victims in an astonishing international...